I couldn't forget about this story for a long time since I restarted my phone. Here's what I can remember. The mc became a character in a novel she usually read and the character isn't the mc of the novel. Better yet, the character was sick and would sometimes throw tantrum to the maids because of her sickness. Knowing the character she possessed is going to die, she did everything she can to stay alive. There was also an episode where she read a letter from her father who supposedly left her, and breakdown crying because the character's memories and hers became one. I don't know how to explain. Oh and the ml of the novel she read was the servant her character apparently abused and became her father's adopted child I think. The father also eventually came to see her after knowing she got a lot better because the mc took control of the character.
This is likely I'm All Out of Health!. It is licensed by Tapas.
Ninenya Seiah’s misfortunes began from birth, owing to her mother’s traitorous reputation and subsequent death. Ninenya was thus banished from the noble Seiah family and forcibly separated from her father, Lord Rubenio Seiah. In extremely poor health, Ninenya has mere months to live. To save his daughter, Rubenio enlists the mysterious Count Adelman as her protector. Ninenya must now work with Adelman to improve her poor health and stop her from becoming the victim of a deadly family feud.
The main character reincarnates as a background character within a novel - this character dies from her sickness, and at her funeral, her father meets the female lead of the original story.
Her sickness is so severe she is perpetually bedridden and can barely eat even the blandest soups. After reincarnating, the main character forces herself to eat soup and medicine in order to force her sickly body to recover.
I'm not sure if this qualifies as a "tantrum". She is rather belligerent with regards to trying to fix her body, but not really in a childish way. There are several such scenes like this, where she tries to force herself in some way and the maids disagree.
The letter scene may be in chapter 19. She reads a simple letter from her father and experiences a sudden outburst of unexpected emotion.
The dialog after the fact is a bit ambiguous as to whether this is the first letter or not. I got the chapter number from a review on anime-planet, not from reading every chapter. There may be a better match for the "character's memories and hers became one" part.